Geometry & Topology, Vol. 6 (2002)
Paper no. 21, pages 609-647.

Boundary curves of surfaces with the 4-plane property

Tao Li

Abstract.
Let M be an orientable and irreducible 3-manifold whose boundary is an
incompressible torus. Suppose that M does not contain any closed
nonperipheral embedded incompressible surfaces. We will show in this
paper that the immersed surfaces in M with the 4-plane property can
realize only finitely many boundary slopes. Moreover, we will show
that only finitely many Dehn fillings of M can yield 3-manifolds with
nonpositive cubings. This gives the first examples of hyperbolic
3-manifolds that cannot admit any nonpositive cubings.